Optimal. Leaf size=138 \[ -\frac {9 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+x \left (c-\frac {c}{a x}\right )^{5/2} \]
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Rubi [A] time = 0.25, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6167, 6133, 25, 514, 375, 98, 154, 156, 63, 208} \[ -\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}-\frac {9 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+x \left (c-\frac {c}{a x}\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 25
Rule 63
Rule 98
Rule 154
Rule 156
Rule 208
Rule 375
Rule 514
Rule 6133
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx\\ &=-\int \frac {\left (c-\frac {c}{a x}\right )^{5/2} (1-a x)}{1+a x} \, dx\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{7/2} x}{1+a x} \, dx}{c}\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{7/2}}{a+\frac {1}{x}} \, dx}{c}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{7/2}}{x^2 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\left (c-\frac {c}{a x}\right )^{5/2} x+\frac {\operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2} \left (\frac {9 c^2}{2}+\frac {c^2 x}{2 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x+\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}} \left (\frac {27 c^3}{4}-\frac {21 c^3 x}{4 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=-\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x+\frac {4 \operatorname {Subst}\left (\int \frac {\frac {27 c^4}{8}-\frac {69 c^4 x}{8 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=-\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x+\frac {\left (9 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}-\frac {\left (16 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x-\left (9 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )+\left (32 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=-\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x-\frac {9 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 116, normalized size = 0.84 \[ \frac {c^2 \left (3 a^2 x^2-26 a x+2\right ) \sqrt {c-\frac {c}{a x}}-27 a c^{5/2} x \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )+48 \sqrt {2} a c^{5/2} x \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{3 a^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 285, normalized size = 2.07 \[ \left [\frac {48 \, \sqrt {2} a c^{\frac {5}{2}} x \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 27 \, a c^{\frac {5}{2}} x \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (3 \, a^{2} c^{2} x^{2} - 26 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a c x - c}{a x}}}{6 \, a^{2} x}, -\frac {48 \, \sqrt {2} a \sqrt {-c} c^{2} x \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - 27 \, a \sqrt {-c} c^{2} x \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (3 \, a^{2} c^{2} x^{2} - 26 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 257, normalized size = 1.86 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \left (-90 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x^{3}+48 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, x^{3}+48 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+45 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{3} a^{2}-48 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{3}-72 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{3} a^{2}-4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{6 x^{2} a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x - 1\right )}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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